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### Paper:

TR10-122 | 18th July 2010 03:22

#### Algorithms for Testing Monomials in Multivariate Polynomials

TR10-122
Authors: Zhixiang Chen, Bin Fu, Yang Liu, Robert Schweller
Publication: 2nd August 2010 05:38
Keywords:

Abstract:

This paper is our second step towards developing a theory of
testing monomials in multivariate polynomials. The central
question is to ask whether a polynomial represented by an
arithmetic circuit has some types of monomials in its sum-product
expansion. The complexity aspects of this problem and its variants
have been investigated in our first paper by Chen and Fu (2010),
laying a foundation for further study. In this paper, we present
two pairs of algorithms. First, we prove that there is a
randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in
an $n$-variate polynomial of degree $k$ represented by an
arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time
algorithm is devised when the circuit is a formula, here $p$ is a
given prime number. Second, we present a deterministic $O^*(2^k)$
time algorithm for testing multilinear monomials in
$\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials, while a
randomized $O^*(1.5^k)$ algorithm is given for these polynomials.
The first algorithm extends the recent work by Koutis (2008) and
Williams (2009) on testing multilinear monomials. Group algebra
is exploited in the algorithm designs, in corporation with the
randomized polynomial identity testing over a finite field by
Agrawal and Biswas (2003), the deterministic noncommunicative
polynomial identity testing by Raz and Shpilka (2005) and the
perfect hashing functions by Chen {\em at el.} (2007). Finally, we
prove that testing some special types of multilinear monomial is
W[1]-hard, giving evidence that testing for specific monomials is
not fixed-parameter tractable.

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